## Andrea Cassinelli (Imperial College London)

**Spectral/hp element methods as a digital twin for turbomachinery applications**

The recent development and increasing integration of high-performance computing, scale resolving CFD and high order unstructured methods offer a potential opportunity to deliver a simulation-based capability (i.e. virtual) for aerodynamic research, analysis and design of industrial relevant problems in the near future. In particular, the tendency towards high order spectral/hp element methods is motivated by their desirable dispersion-diffusion properties, that are combined to accuracy and flexibility for complex geometries.

The effects of polynomial order, spanwise domain extent and spanwise resolution are explored in a T106A vane with clean inflow boundary conditions. However, realistic turbomachinery flows are highly turbulent. Building on such experiments, we therefore analyze the performance of a representative industrial cascade at moderate Reynolds number with various levels and types of inflow disturbances. The introduction of a steady/unsteady spanwise-nonuniform momentum forcing in the leading edge region is explored, to break the flow symmetry upstream of the blade and investigate the change in transition mechanism in the aft portion of the suction surface. To provide a systematic turbulence generation tool, a synthetic inflow turbulence generation method is incorporated and applied to a representative industrial low pressure turbine vane. The clean results of the cascade are compared to various levels of momentum forcing and inflow turbulence, looking at blade wall distributions, wake profiles and boundary layer parameters.

Low levels of background disturbances are found to improve the agreement with experimental data. The results support the confidence for using high order spectral methods as a standalone performance analysis tool but, at the same time, underline the sensitivity at these flow regimes to disturbances or instabilities in the real environment when comparing to rig data.

## Giacomo Castiglioni (Imperial College London)

**Shock-Wave Boundary Layer Interactions Simulations with a Discontinuous Spectral/hp Element Method**

Shock wave boundary layer interaction (SWBLI) is a phenomena encountered in many industrial devices such as transonic and supersonic external aerodynamic applications, supersonic air intakes, transonic and supersonic cascades, and over-expanded nozzles. SWBLI plays a critical role to the design of such devices due to its importance for both efficiency and structural integrity, often being the limiting factor to the design envelope. Although SWBLIs have been extensively (Dolling, 2001) there are still many open questions and they remain a very active research area. Of particular interest are turbulent SWBLIs and SWBLIs in non-trivial geometries. Lately discontinuous Galerkin spectral element methods (DGSEM) have gained popularity for the solution of the turbulent compressible flows due to their spectral accuracy, geometrical flexibility, and scalability. These characteristics make of DGSEM a promising candidate to be the platform for a new generation of computational fluid dynamics software. Several different approaches are proposed in literature to tackle shocks in conjunction with DGSEM including limiters, artificial viscosity, and filtering procedures. On the other hand, it is noteworthy that so far most methods have been applied only to two-dimensional inviscid flows, see brief overview in Chaudhuri et al. (2017).

The aim of the present work is to evaluate different shock capturing strategies encompassing different shock sensors and different strategies for adding artificial viscosity, for instance augmenting the right hand side of the Navier-Stokes equations with a Laplacian term augmenting the physical viscosity and thermal conductivity. The long term objective is to improve the efficiency, and robustness of the existing solver in order to tackle turbulent SWBLIs in industrially relevant geometries. As a test case for SWBLI we selected a laminar SWBLI problem which has been extensively studied experimentally and numerically.

## Vishal Saini (Loughborough University)

**Performance Comparison of Standard Finite-Volume and Spectral-hp Methods for the LES of Representative Gas Turbine Combustor Aerodynamics**

An evaluation of computational cost versus accuracy for Large-Eddy Simulations (LES) is performed using standard second-order and high-order accurate unsteady incompressible solvers from OpenFoam and Nektar++ frameworks respectively. The focus of the present study is on coarse simulations considering the industrial demands. The subgrid scales of the flow are treated using WALE model [1] in OpenFoam and spectral vanishing viscosity (SVV) [2] in Nektar++ simulations. To begin with, the 3D Taylor-Green vortex case is considered using fully hexahedral and tetrahedral meshes. It is found that, for a given level of accuracy, the incompressible spectral-hp simulations are 2 to 4 times cheaper than the low-order counterpart. Motivated by this outcome, a similar comparison is undertaken on a rich-burn gas turbine combustor representative case, studied experimentally by Spencer et al. [3]. The flow consists of six radial jets impinging on main cross-flow resulting in high turbulence levels, which closely resembles the dilution port flows of a combustor. For the first part of the performance comparison, the computational time of the low- and high-order simulations is matched by differing the meshes sizes (while keeping a similar distribution of the solution points). The initial analysis of the time spectra (of velocity signals) at key spatial locations suggests that the spectral-hp solver resolves a relatively larger range of flow scales.

*References*

[1] F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 1999.

[2] R. M. Kirby and S. J. Sherwin. Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling. Computer Methods in Applied Mechanics and Engineering, 2006.

[3] A. Spencer and J.J. McGuirk. LDA measurements of feed annulus effects on combustor liner port flows. Journal of fluids engineering, 2001

## Stanislaw Gepner (Technical University of Warsaw)

**From Hydrodynamic instability to chaotic mixing**

Improvements in the performance of various flow-based devices, such oxygenators or DNA microarrays can be reduced to decreasing hydraulic drag and increasing achievable mixing efficiency. In case of fluids, mixing can be seen as an inter-material transport process (such as diffusion) overlaid on top of “mechanical” stirring that results from kinematics of the flow. Since, in general diffusion is relatively slow, the role of stirring is to cause generation of small scale structures that can be quickly smoothed out by diffusion. This can be efficiently achieved via turbulization. Still, there are a few problems for which this is unfeasible, either due to size constraints or fragile nature of processed materials. In case of laminar flows effective stirring can be achieved by principles of chaotic advection. Chaotic advection is a phenomenon in which fully laminar, relatively simple velocity fields (in the Eulerian representation) result in chaotic response in the Lagrangian view. High complexity of motion, in case of such flows, results from the fact that dynamical system that describes motion of individual fluid particles might be non-integrable leading to the onset of chaotic trajectories and in consequence improved stirring. The objective of this work is to study the onset of chaotic advection due to amplification of naturally occurring hydrodynamic instabilities resulting from large-scale wall corrugations. At the same time we plan to examine and quantify possible improvements of mixing due to chaotic character of advection.

## Hongyi Jiang (University of Western Australia)

**Use of Nektar++ and OpenFOAM for the simulation of bluff-body flows**

Nektar++ and OpenFOAM are two popular computational fluid dynamics (CFD) codes for research purposes. Based on the use of both codes for the past four years, I will talk about some advantages of Nektar++ over OpenFOAM for the simulation of bluff-body flows. First, Nektar++ uses Fourier expansion in the spanwise direction of the cylinder, in comparison with the replication of mesh elements for the conventional finite element/volume method such as OpenFOAM. For the latter, the skewness of the mesh elements in the cross-sectional plane would induce persistent numerical disturbance in the spanwise direction, which would result in a weak artificial three dimensionality of the flow and may affect the judgment of flow physics. The second advantage of Nektar++ is a higher accuracy offered by the spectral/hp element method, which is extremely useful for the study of highly sensitive (mesh-dependent) flows such as the secondary vortex street in the far wake of the cylinder.

Besides the advantages, some potential improvements are also suggested. First, since the high-order data from Nektar++ are imported into visualisation packages as linear approximations, it would be useful to have a FieldConvert utility to generate an interpolated flow field with higher precision than linear approximation for visualisation and post-processing purposes. Second, it would be more efficient for the visualisations if the flow fields and mesh are stored in separate files, especially for the three-dimensional cases where the file sizes are large.

## Walid Hambli (Imperial College London)

**High-order methods for Automotive and Racing Applications**

The automotive and motor racing industries are among the most demanding in terms of CFD simulations. Currently road and race vehicles designs have very complex geometries and offer great challenges to predict the aerodynamic behaviour due to high Reynolds numbers, lower ground height, flow separation and wake profile. Predicting the correct flow physics is the key factor of improving vehicle aerodynamic performance and high-fidelity methods are becoming a trend. We present the latest work with the spectral/hp element method on Nektar++ for automotive and motor racing applications using an implicit LES simulation with novel spectral vanish viscosity (SVV) DGKernel stabilization technique. We start presenting methodology and correlation studies on the Ahmed Body and further application of underbody diffuser on the same body. Moving to a real vehicle component, open and closed wheel studies are presented for different polynomial expansion. We move further with the aerodynamic study on the Elemental car full vehicle and finish with the correlation study on a Formula One front wing in cooperation with McLaren Racing. Cases here presented indicate the capacity of the methodology and the code to handle complex industrial geometries and provide accurate results.

## Rodrigo Moura (Instituto Tecnológico de Aeronáutica)

**Under-resolved DNS of non-trivial turbulent boundary layers via spectral/ hp continous Galerkin methods**

We present results on the suitability of spectral/hp continuous Galerkin (CG) schemes for model-free under- resolved simulations of non-trivial turbulent boundary layer flows. We consider a model problem proposed by P. R. Spalart that features a rotating free-stream velocity and admits an asymptotic solution with significant cross-flow effects. This test case is substantially more complex than typical turbulent boundary layer canonical problems owing to its unsteadiness and enhanced small-scale anisotropy. Reported LES- based solutions to this problem are known to require sophisticated modelling and relatively fine grids to achieve meaningful results, with traditional models exhibiting poor performance. The model-free CG-based approach advocated, on the other hand, yields surprisingly good results with considerably less degrees of freedom for higher-order discretisations. Usefully accurate results for the mean flow quantities could even be obtained with half as many degrees of freedom per direction (in comparison to reference LES solutions). Usage of high-order spectral element methods (CG in particular) is therefore strongly motivated for wall- bounded turbulence simulations via under-resolved DNS (uDNS), sometimes called implicit LES (iLES), approaches.

## Abhishek Kumar (Coventry University)

**Stability of stratified fluid in a nearly semicircular pool**

In this work, we investigate the stability of stably stratified flow in a nearly semicircular pool with an upper free surface where fluid can be fed in, and with porous lower boundaries where fluid can escape. This generic geometry is representative of numerous problems were solid materials are melted, as for example in metallurgical casting processes [1]. We solve the equations governing stably stratified flow under Boussinesq approximation in a curved geometry using the spectral element code Nektar++ [2]. The linear stability analysis shows that the two-dimensional steady base flow becomes unstable to oscillatory or non-oscillatory three-dimensional (3D) modes depending on the inlet net mass flux. Further, the analysis of the nonlinear evolution of the system using 3D direct numerical simulation and the Stuart—Landau equation [3] exhibit both supercritical as well as subcritical transition for different parameters.

*References*

[1] Flood and Davidson, Mat. Sci. Tech., (1994).

[2] Cantwell et al., Comput. Phys. Commun., (2015).

[3] Sheard et al., J. Fluid Mech., (2004).

## Sehun Chun (Yonsei University)

**Aligning Frames along a Connection to draw the Atlas**

We often solve various PDEs in various domains with many parameters, but we sometimes miss the point of what the meaning of PDEs is. Why is the wave that the PDEs model so important? and which properties are we going to regenerate by solving the equations. One proposition on the purpose of PDEs is to construct a ‘connection’ between the local points. This connection becomes a ‘curvature’, meaning a mass or energy in sense of physics or biology, to create a force along the line. If we know the connection and curvature the propagation, we know the ‘Atlas’ of the propagation that is all we need to know for the understanding of some interesting phenomena in space, and even time. In this talk, I explain how we construct this Atlas in the context of Nektar++ and how it can be used in the fields of cardiology and neuroscience for improving analysis and prediction of related diseases.

## Joaquim Peiró (Imperial College London)

**NekMesh:**** An open-source high-order mesh generator**The generation of suitable, good quality high-order meshes is a significant obstacle in the academic and industrial uptake of high-order CFD methods. We will present the recently released high-order mesh generator

*NekMesh*, which is part of the open-source

*Nektar++*spectral/

*hp*element framework project (www.nektar.info) and generates full tetrahedral or mixed prismatic and tetrahedral high-order meshes from existing linear meshes using a pipeline approach.

*NekMesh*is a high-order mesh generation and modification program which can: convert meshes from one format to another; edit meshes whilst retaining high-order information; and generate high-order meshes from a CAD definition. It is set up in pipeline style where a series of modules are constructed and executed in order to arrive at a high-order mesh. A typical execution will involve an input stage, some processing (mesh editing) stages if required and an output. This arrangement means that the program can easily be used for a wide number of applications and can, with relative ease, be set up to read or write the high-order mesh in a number of different formats. The modular structure of

*NekMesh*makes it easy to incorporate different third-party CAD engines and pre- and post-processors into the process. At the workshop we will describe in some detail the various stages involved in the

*NekMesh*pipeline for the generation of high-order meshes from a CAD description.

## Jeremy Cohen (Imperial College London)

**Tools to support usability and training for Nektar++**

Advanced computational codes such as Nektar++ continue to be increasingly widely used in research. As the user base of Nektar++ grows, many newcomers to the software are inexperienced with working with large-scale, specialist research software. They may have limited experience with building code from source or interacting with software via the command line. In this talk I will look at three different elements of ongoing work to develop tools and services that help make Nektar++ more accessible to a wider group of potential users. The talk will first look at developments with the Nekkloud web-based tool to provide a meshing console that supports generation of 2D and 3D meshes using NekMesh via a web-based user interface. I will then look at a tool that provides straightforward access to Nektar++ containers that be easily downloaded and deployed locally on a user’s system or on a remote cluster. I will conclude with an overview of some initial investigations that have been undertaken into building a version of Nektar++ that can run within a web browser to support demos and tutorials.

## Edward Laughton (University of Exeter)

**Non-conformal mesh interfaces in 2D with the discontinuous Galerkin method**

Traditionally Nektar++ has utilised purely conformal meshes where the intersection between any two elements consists of a shared sub-element. An end goal of allowing for problems with moving geometry is the motivation for expanding the code to also handle non-conformal meshing. This presentation will outline progress along these lines, highlighting a working prototype for non-conformal 2D discontinuous Galerkin simulations and enabling the use of non-conformal meshes in, for example, the Advection-Diffusion-Reaction and compressible Navier-Stokes solvers. The code presently has the ability to split any provided mesh along an interface or work with a pre-split mesh. Initial results demonstrate that the high-order convergence properties are unaffected, with a minimum increase in computation time due to the initial interpolation along the interface. This code also makes use of additional bounding-box optimisations to facilitate fast element searches, changes to how domains are handled and a Lagrange interpolation speedup using the barycentric formulation.

## Jan Pech (Czech Academy of Sciences)

**Temperature dependent material properties in incompressible flow solver**Numerical simulations of non-stationary flows influenced by heating/cooling are demanded in both research and industrial applications (e.g. heat exchangers, energy transfer, change in flow structures due to temperature change). One of the mathematical models in this problematics consists of fully coupled momentum balance and strongly non-linear energy equation. The talk will focus to the solution strategy for the Navier-Stokes-Fourier system with temperature dependent material properties, whose 2D version was already implemented in Nektar++ framework and is supposed to become a part of a future Nektar++ release. Solver developed as an extension of the velocity-correction scheme used in the Incompressible Navier-Stokes solver of Nektar++ allows nontrivial velocity divergence, which occurs as a consequence of modelling the thermal expansion while direct dependence of density on pressure is neglected. Our approach allows viscosity, density and thermal conductivity to be temperature dependent without any a priory restriction to the form of these dependencies. Verification on manufactured solutions as well as results from applications in 2D including various effects as buoyancy or viscous heating will be presented.

## Zhenguo Yan (Imperial College London)

**Development of implicit compressible flow solver in Nektar++**

To accelerate the DGSEM simulations of compressible flows in the open source package Nektar++, implicit time integration methods are developed. The long term objective is to develop efficient time integration methods for large scale high-fidelity simulation of turbo-machinery. To achieve this objective, the multi-stage diagonally implicit Runge-Kutta method is used to descretize the time derivatives and the Jacobian-free Newton Krylov method is used to solve the nonlinear systems in each implicit stage. The Jacobian-free method is adopted to avoid explicitly calculating the large Jacobian matrix. Different preconditioners for the generalized minimal residual method (GMRES), the specific Krylov method adopted, are developed, which includes the block Jacobi preconditioner and the Incomplete LU factorization preconditioner. The influences of different preconditioners and different approximations of the Jacobian matrix on the convergence efficiency and memory consumption are investigated, the understanding of which is important for developing new preconditioners to further improve efficiency and reduce footprint. Based on the current implicit solver, computation efficiency and accuracy are compared with the existing explicit solver on typical test cases, such as the vortex shedding from a sub-sonic cylinder. The results have shown that obvious acceleration can be achieved while maintaining good accuracy.

## Charles Houston (Imperial College London)

**Simulating the electrophysiology of discretely-coupled cardiac cells in a multi-domain formulation**

*Background*

Simulations of action potential propagation in cardiac tissue are increasingly being used to understand driving mechanisms in clinically-driven translational cardiac electrophysiology. These simulations generally model cardiac tissue as a functionally homogeneous medium, which is a physiological inconsistency with the heterogeneous microstructure of the heart. To investigate the genesis of key arrhythmia mechanisms requires these cellular components to be modelled explicitly. We aim to build a biophysically accurate ‘discrete-cell’ model of cultured cardiac cells which can predict observed microscopic conduction patterns.

*Methods*

The model is implemented in the Nektar++ spectral/hp-element method framework, with multiple subdomains representing individual cells and a separate subdomain representing the extracellular space. Subdomains are assumed to be volume conductors with no internal current sources. Cell subdomains are coupled to each other through gap junction interfaces, and to the extracellular subdomain through membrane interfaces. Gap junctions are modelled as ohmic resistors and membranes have a capacitive current and ionic current (calculated from the output of a model of a cardiac cell action potential). Two solutions exist on interfaces (one in either opposing subdomain). Current flow continuity between subdomains is enforced directly in the matrix system, with coefficients at coupled interfaces ordered first in the solution vector to reduce the computational cost of the solve step by taking advantage of orthogonality between subdomains.

*Results & Discussion*

Two known analytical solutions for a cable of cardiac cells under different conditions are used for initial validation of the model.The simulation faithfully reproduces both an analytical solution of constant subthreshold potential maintained at the end of the cable of cells, and the solution for a current injected into the cable end. Increasing gap junctional resistance is shown to increase the magnitude of discrete jumps in intracellular potential across interfaces between cells.

*Conclusion*

A multi-domain formulation to simulate discretely-coupled cardiac cells has been implemented using spectral/hp-element methods. The model will continue to be developed through biophysical validation against action potential propagation in biological preparations.